We study the 3-parameter family \(T_{t, \theta }\cap S_s\) , \((t, s, \theta ) \in [-1, 1] \times [-1, 1] \times [0, 2\pi ] /\)\(0 \equiv 2\pi [-1, 1] \times [-1, 1] \times S^1\). These intersections may be regarded as the level set at height = \(s\)  of a 2-parameter family of functions on \(T\) . All told this means we are looking for a 3-parameter family of germs of smooth functions \((R^2, 0) \rightarrow (R, 0)\) . According to Thom's Jet-transversality Theorem any such family can be perturbed a generic one with local simulation types as listed below. In these dimensions the generic local germs All other germ types such as hyperbolic umbilic, \(f(x, y) = x^3 + y^3\) , have codimensions \(\ge 4\)  and need not be considered. [1]} are represented by:
<TABLE>These are the germs. We also may invoke genericity to ensure that where a single function on \(T\)  restricts to singular germs on several different points of \(T\) , the local unfoldings are transverse. These is no difficulty making this precise as the stratification of the unfoldings obey the two Whitney conditions [2]} are are in fact piecewise smooth submanifolds of the parameter space.
So what does the singular set of a graphic in \([-1, 1] \times [-1, 1] \times S^1\)  look like? It consists of 2D-manifold sheets of two types - saddle and sink - meeting along birth/death 1D strata, which are allowed to have dovetail singularities at isolated points. The 1D and 2D strata may pass transversely through each other at points and the 2D strata may have transverse 1-manifolds of intersection and isolated standard triple points.
The picture we must study is much simpler. Just as in [3]}, the 2D sheets labeled by source/sink tangencies may be discarded forming the reduced graphic, as crossing these walls only changes the intersection \(T \cap S\)  by inessential simple closed curves. The remaining saddle-labeled walls (it does not matter for connectivities, but for convenience take their closures) divide \([-1, 1] \times [-1, 1] \times S^1\)  into complementary open regions \(\lbrace R_i\rbrace \) , each of which has a constant topological pattern \(T \cap S\) , up to birth and death of inessential sccs. Thus the 3D reduced graphic \(G_3\)  is a straightforward generalization of the 2D one: 2-manifold sheets (perhaps with borders) crossing in double curves and triple points.
The graphic on an annular surface
After having defined the 3D reduced graphic, we now replace it with a 2D graphic by restricting it to the sub-surface \(\Sigma \subset [-1, 1] \times [-1, 1] \times S^1\) , where \(\Sigma \)  is the graph of the function described in Proposition REF . Its structure as a graph provides \(\Sigma \)  with a natural diffeomorphism to an annulus \([-1, 1] \times S^1\) , parameterized by \((t, \theta )\) , and that is how we will view it. This structure as a graph allows us, when thinking of the intersection \(T_{t, \theta }\cap S_s\)  that corresponds to a point in \(\Sigma \) , to take the sphere as fixed, so henceforth we drop the subscript \(s\) . (When there is little risk of confusion, we will also drop the subscript on \(T\) .) Each region of the graphic in \(\Sigma \)  corresponds to a transverse intersection of \(T_{t, \theta }\)  with \(S\)  in which some of the curves of intersection are essential in \(T_{t, \theta }\)  and bound disks in \(A_{t, \theta }\)  or \(B_{t, \theta }\) , perhaps both. For regions near \(t = -1\) , all such disks lie in \(A_{t, \theta }\)  (since \(T\)  is near the spine of \(A_{t, \theta }\) , which we may take transverse to the height function \(s\) ) and we label these regions \(A\) . Similarly, near \(t = +1\) , all such disks lie in \(B_{t, \theta }\)  and we label such regions \(B\) .
Let us establish the following labeling and “artistic" convention for the general region \(R\)  of our annular graphic. If \(T_{t, \theta }\cap S\)  contains a scc \(a\)  essential in \(T_{t, \theta }\)  and compressing in the handlebody \(A_{t, \theta }\)  label the region containing \((t, \theta )\)  by \(A\) . (Do this regardless of whether \(T_{t, \theta }\cap S\)  contains an essential scc compressing in \(B_{t, \theta }\) ). If \(T_{t, \theta }\cap S\)  does not contain any essential scc compressing in \(A_{t, \theta }\) , but does contain a scc compressing in \(B\) , label that Region \(B\) .
Artistic convention: If labels \(A\)  and \(B\)  alternate around a saddle wall double point render the boundary between \(A\) -regions and \(B\) -regions to be a 1-manifold favoring \(A\) . We label this 1-manifold \(M\) , it separates \(A\)  regions from \(B\)  regions. We recall both this labeling and rendering convention with the phrase: “favor \(A\) ."
Since near one boundary component of the annulus \(\Sigma \)  the regions are all labelled \(A\)  and at the other they are all labelled \(B\) , there is at least one component of \(M\)  that is essential in \(\Sigma \) . Such a component will be homotopic in \(\Sigma \) , fixing a point, to a core curve \(\lbrace pt\rbrace  \times S^1 \subset [-1, 1] \times S^1\) . Here then is the plan for the proof of Theorem REF . We will show

Each component of \(M\)  in \(\Sigma \)  naturally describes an edge-path in \(2C(T)\)  through vertices \((a, b)\)  in which the first curve bounds an essential disk in \(A\)  and the second an essential disk in \(B\)

For any component of \(M\)  that is essential in \(\Sigma \)  the associated edge-path is a cwr representing \(\tau \) .

<FIGURE>The crux is to understand the intersection locus where two saddle walls cross, what the four possible resolutions of the two saddles look like, and which curves among these resolution could be labeled \(a\)  or \(b\) . (Again a curve is labeled \(a\)  (resp \(b\) ) if it is essential in \(T\)  and compresses in \(A\)  (resp \(B\) ).)
The singular pattern \(\xi \)  associated with a saddle wall crossing is present simultaneously in the sphere \(S\)  and the surface \(T\) . Thus the pattern \(\xi \)  as a 4-valent graph is the same in \(S\)  and \(T\) . Furthermore the tangential information is also preserved so at a saddle tangency we know which legs are opposite (this is not quite the information of a cyclic ordering; it might be called a "dihedral ordering" as the information is a coset of \(S(4) / D(4)\) ). But interestingly, it is not true that the pairs \((\mathcal {N}_S(\xi ), \xi ) \subset S\)  and \((\mathcal {N}_T(\xi ), \xi ) \subset T\)  are necessarily homeomorphic. (Here \(\mathcal {N}_S\)  and \(\mathcal {N}_T\)  denote neighborhood in \(S\)  and \(T\)  respectively.) The important example is Figure REF ; it is a torus of revolution intersected with a sloping plane to produce two circles of intersection. In the plane they have one positive and one negative intersection, whereas in the torus they have two positive intersections.
<FIGURE>For our analysis we need to list all possible pairs \((\mathcal {N}(\xi ), \xi )\)  up to homeomorphism. Let us start by enumerating the possibilities on \(S\) . There are 4:
<FIGURE>These four possible \(\xi \) s can yield neighborhood pairs on \(T\)  as shown in Figure REF . (The small circles in the figures represent boundary components of the neighborhood, typically essential circles in \(T\) .)
<FIGURE>So, overall, on \(T\)  we have 5 cases to consider: 1, 2, 3, 4, and 5 above.
Figure REF  displays the four local resolutions within \(\mathcal {N}(\xi )\)  of \(\xi \)  in each of the 5 cases.
<FIGURE>Most of \(M \subset \Sigma \)  runs along arcs in the graphic, representing saddle walls for which one side is labelled \(A\)  and the other is labelled \(B\) . A neighborhood of the corresponding saddle is a 3-punctured sphere and crossing the saddle wall splits one curve into two: \(c \rightarrow c^\prime \mathbin {\rotatebox {180}{\XMLaddatt {origin}{c}\Pi }}c^{\prime \prime }\) .
We start with a simple lemma.

Lemma 4.3 (two of three)  
If crossing a saddle wall transforms \(c \rightarrow c^\prime \mathbin {\rotatebox {180}{\XMLaddatt {origin}{c}\Pi }}c^{\prime \prime }\)  and if two of the three curves compress in \(A\)  (resp \(B\) ) then so does the third.

Consider capping off two boundaries of the 3-punctured sphere.
Of course the third curve may be inessential in \(T\) , so crossing from a region labelled \(A\)  to one labelled \(B\)  may represent two curves bounding essential disks in \(A\)  fusing at the saddle into a curve inessential in \(T\) , leaving other unrelated curves bounding disks in \(B\) . (In this case the two fused curves at \(A\)  are isotopic in \(T\) .) Or passing through could represent a curve that bounds a disk in \(A\)  fusing with a curve that is essential in both \(A\)  and \(B\)  to create a curve bounding a disk in \(B\) , or a similar fission. In each case there is a unique essential intersection curve \(a\)  bounding a disk in \(A\)  associated with the saddle wall and it is disjoint from every essential intersection curve bounding a disk in \(B\) .
To expedite the discussion we introduce:

Definition 4.4
Some conventions:
We call a scc \(a\)  (resp \(b\) ) if it is essential in \(T\)  and compresses in \(A\)  (resp \(B\) ).

An edge-path in \(2C(T)\)  in which each vertex is of type \((a, b)\)  (as just defined) is an admissible path.

Near a saddle wall crossing (swc) point \(p\)  we call a scc of type \(a\)  or \(b\)  local if it is a scc of one of four resolutions of \(\xi _P\) . Otherwise we call it far.


With these conventions, we have just observed that any saddle wall that is part of \(M\)  is associated to a `cloud' of vertices in \(2C(T)\)  of the form \((a, b)\) , with \(a\)  uniquely defined, and \(b\)  any curve of that type coming from the adjacent graphic region labelled \(B\) . Any two vertices in such a cloud are connected via the relation
\((a, b) \rightarrow (a, b^\prime ) \: \textrm {since \) 
\(, \) b\(, \) b\( are all pairwise disjoint.}\)\(\)
Recalling our artistic rendering convention for defining the 1-manifold \(M\) , we now consider how these clouds of vertices in \(2C(T)\)  are related as we pass from one saddle wall to another, either through or around a corner of a vertex in the graphic representing a saddle wall crossing.

Lemma 4.5 For all cases (i)
\(\begin{array}{c|c}A & A \\\hline B & B\end{array}\) 
, (ii)
\(\begin{array}{c|c}A & B \\\hline B & A\end{array}\) 
, (iii)
\(\begin{array}{c|c}A & A \\\hline A & B\end{array}\) 
, and (iv)
\(\begin{array}{c|c}A & B \\\hline B & B\end{array}\) 
there is an admissible path in \(2C(T)\)  from some vertex in one cloud to some vertex in the other.

Since each pair of vertices in a cloud are connected by an edge in \(2C(T)\) , and such an edge is obviously an admissible path, it suffices to replace both occurences of `some' with `every' in its two occurences in the lemma.
Case (i)
\(\begin{array}{c|c}A & A^{\prime } \\\hline B & B^{\prime }\end{array}\) 
: If any \(b\)  (or any \(b^\prime \) ) is far it will be unaffected by passing through any saddle wall, so pick \(b = b^\prime =\)  far. Moreover, passing through the saddle wall between the regions \(A\)  and \(A^{\prime }\)  will change the \(a\)  curve to a disjoint \(a^{\prime }\) . Then \((a, b) \rightarrow (a^\prime , b) = (a^\prime , b^\prime )\)  is an admissible path from one cloud to the other.
If \(b\)  and \(b^\prime \)  are both local there is a subtlety. In all cases at most one diagonal pair of quadrants contain intersecting curves (see Figure REF ); call this a dangerous diagonal. But the appropriate choice of a two-step process avoids the dangerous diagonal, so all curves are disjoint and the path is admissible:

[baseline=([yshift=9pt]current bounding box.south)]
t (0,1) \((a, b)\) ;
[->] (0.6, 1) – (1.6, 1.4);
[->] (0.6, 1) – (1.6, 0.6);
right] at (1.6, 1.4) \((a, b^\prime )\) ;
right] at (1.6, 0.6) \((a^\prime , b)\) ;
right] at (1.92, 1) or;
[->] (2.8, 1.4) – (3.8, 1.1);
[->] (2.8, 0.6) – (3.8, 0.9);
right] at (3.8, 1) \((a^\prime , b^\prime )\) ;
Case (ii)

[baseline=([yshift=9pt]current bounding box.south)]
[rounded corners] (0, 0.4) – (0.4, 0.4) – (0.4, 0);
[rounded corners] (0.45, 0.85) – (0.45, 0.45) – (0.85, 0.45);
above right] at (-0.1, -0.1) B;
above right] at (-0.1, 0.4) A;
above right] at (0.4, 0.4) B;
above right] at (0.38, -0.1) A;

: It suffices to consider one arc of \(M\) . Regardless of whether \(b\)  is local or far leave it fixed. The only problematic possibility for a transition \((a, b) \rightarrow (a^\prime , b)\)  is if \(a\)  and \(a^\prime \)  are intersecting curves from a dangerous diagonal. In this context case (5) cannot occur since there \(a\)  and \(a^\prime \)  would have intersection number = 1. But where it is possible, cases (2), (3), and (4), the topology is the same: \(a\)  and \(a^\prime \)  are crossing non-boundary-parallel sccs on a 4-punctured sphere:
<FIGURE>But an innermost circle argument reduces the geometry to precisely the case pictured in Figure REF , where we see that all four boundary curves also compress in \(A\) . By Lemma REF  at most two of these boundary curves are inessential in \(T\) ; let \(\partial \)  be a boundary curve that is essential in \(T\) .
Then \((a, b) \rightarrow (\partial , b) \rightarrow (a^\prime , b)\)  is an admissible path from one cloud to the other. (Note that the intermediate vertex \((\partial , b)\)  is not in either cloud.) Call this the lantern trick.
Case (iii)

[baseline=([yshift=9pt]current bounding box.south)]
[rounded corners] (0.45, 0) – (0.45, 0.45) – (0.85, 0.45);
above right] at (-0.1, -0.1) A;
above right] at (-0.1, 0.4) A\(\prime \);
above right] at (0.37, 0.4) A\(\prime \prime \);
above right] at (0.4, -0.1) B;

: This is similar to case (ii). Regardless of whether \(b\)  is local or far, leave it fixed. The only problematic case in passing from \((a, b)\)  to \((a^{\prime \prime }, b)\)  is if \(a\)  and \(a^{\prime \prime }\)  are intersecting sccs from a dangerous diagonal. As in case (ii), case (5) cannot occur. Cases (2), (3), and (4) are also handled as in case (iii) by the lantern trick.
Case (iv)

[baseline=([yshift=9pt]current bounding box.south)]
[rounded corners] (0, 0.43) – (0.42, 0.43) – (0.42, 0.85);
above right] at (-0.07, -0.1) B;
above right] at (-0.1, 0.4) A;
above right] at (0.37, 0.4) B\(\prime \prime \);
above right] at (0.37, -0.1) B\(\prime \);

: If any \(b\) , \(b^\prime \) , \(b^{\prime \prime }\)  is far, select it and keep it constant, and keep \(a\)  constant as well. Then the corresponding vertex in \(2C(T)\)  lies in both clouds
Now assume all \(b\) , \(b^\prime \) , \(b^{\prime \prime }\)  are local.
Case (1): If \(a, b, b^{\prime }, b^{\prime \prime }\)  are all disjoint, then \((a, b) \rightarrow (a, b^{\prime \prime })\)  is an admissible path.
Case (2), (3), (4): The only non-disjoint situation is when \(b\)  and \(b^{\prime \prime }\)  are intersecting curves from a dangerous diagonal pair. Again use the lantern trick.
Case (5): This case cannot occur: By intersection number, the three \(B\)  regions cannot contain a dangerous diagonal pair, so \(A\)  must be one of the dangerous diagonal pair of quadrants. Then the other quadrant in that pair must be \(B^\prime \) . The other two quadrants then are represented by the boundary curves in the twice-punctured torus shown in Figure REF , one compressible in \(A\)  and the other in \(B\) . In order to avoid a label \(A\)  in one of these quadrants, the former curve must be inessential in \(T\)  so the twice-punctured torus is actually only once-punctured. But in that case, the label \(B\)  (or \(B^{\prime \prime }\) ) for this quadrant implies that there is a (far) \(b\)  curve, contradicting assumption.
We now complete the proof of Theorem REF  as described in the plan above. Pick an essential component of the 1-manifold \(M\)  in \(\Sigma \)  and a saddle wall edge in it. Beginning at that edge and moving around the component of \(M\) , construct an admissible path through \(2C(T)\)  as described above, until one returns to the original edge. Since the component is homotopic in \(I \times I \times S^1\) , rel its beginning and ending point, to a simple loop \(\lbrace pt\rbrace  \times S^1\)  which represents \(\tau \) , the cwr given by the admissible path also represents \(\tau \) .

The genus 3 case
We are now in a position to prove the Powell Conjecture for genus 3 splittings of \(S^3\) , by verifying Conjecture REF  in this case.
We need a bit of terminology. Suppose \(H\)  is a genus 3 handlebody and \(D \subset H\)  is a separating disk. Then \(D\)  divides \(H\)  into a solid torus \(H_D\)  and a genus 2 handlebody; the meridian disk \(D^{\prime }\)  of \(H_D\)  is well-defined up to proper isotopy in \(H\)  and is non-separating in \(H\) ; call \(D^{\prime }\)  the surrogate of \(D\)  in \(H\) .
Lemma 5.1 Suppose \(S^3 = A \cup _T B\)  is a genus 3 splitting, and disks \(a \subset A, b \subset B\)  are a weakly reducing pair. Then at least one of the following 4 disks is primitive:

\(a\) , if \(a\)  is non-separating in \(A\) ,

the surrogate of \(a\)  if \(a\)  is separating in \(A\)

\(b\) , if \(b\)  is non-separating in \(B\)

the surrogate of \(b\)  if \(b\)  is separating in \(B\)


Case 1: \(a\)  is separating. One possibility in this case is that \(\mbox{$\partial $}b\)  lies on the boundary of the solid torus \(A_a \subset A\) . In that case, if \(b\)  is non-separating, then \(\mbox{$\partial $}b\)  must be a longitude of \(A_a\) , verifying that the surrogate of \(a\)  (and incidentally also \(b\) ) is primitive. If \(b\)  is separating, then \(\mbox{$\partial $}b\)  must be parallel to \(\mbox{$\partial $}a\)  in \(T\) , so together \(a\)  and \(b\)  constitute a reducing sphere for \(T\) , cutting off a genus 1 Heegaard splitting whose splitting surface is \(\mbox{$\partial $}A_a\) . It follows again that the surrogate of \(a\)  (i. e. the meridian of \(A_a\) ) is primitive (and incidentally that the surrogate of \(b\)  is also primitive).
The other possibility is that \(\mbox{$\partial $}b\)  is essential in the punctured genus 2 component of \(T - \mbox{$\partial $}a\) . This implies that the surrogate of \(b\)  (if \(b\)  is separating) is disjoint from the surrogate of \(b\) , so we may as well jump immediately to Case 2.
Case 2: Both \(a\)  and \(b\)  are non-separating. Let \(T^{\prime \prime } \subset S^3\)  be the torus obtained by simultaneously compressing \(T\)  along \(a\)  and \(b\) . \(T^{\prime \prime }\)  bounds a solid torus \(W\)  on one side or the other, say \(W\)  lies on the side in which \(b\)  lies. The surface \(T^{\prime }\)  obtained from \(T\)  by compressing only along \(a\) , when pushed into \(W\) , describes a genus 2 Heegaard splitting \(W = A^{\prime } \cup _{T^{\prime }} B^{\prime }\)  in which \(\mbox{$\partial $}W = \mbox{$\partial $}_- B^{\prime }\)  and \(b\)  is the unique boundary-reducing disk for the compression body \(B^{\prime }\) . It follows that \(b\)  is primitive in \(T^{\prime }\) , hence in \(T\) .
Following Lemma REF  and Corollary REF  there is an obvious candidate for assigning a Powell equivalence class of homeomorphisms to the pair \((a, b)\) : if \(a\)  (or its surrogate) is primitive, assign the class of homeomorphisms that carries that disk to the disk \(a_1\)  in the standard splitting \(A \cup _{T_3} B\) . If \(b\)  (or its surrogate) is primitive, assign the class of homeomorphisms that carries that disk to \(b_3\)  in the standard splitting. It is only necessary to check that if two of the disks listed in Lemma REF  are primitive, the assignments we have made coincide. So suppose both \(a\)  (or its surrogate) and \(b\)  (or its surrogate) are primitive.
If both \(a\)  and \(b\)  are non-separating (so surrogates are not involved) and therefore primitive, a straightforward outermost arc argument will find a disk \(b_{\perp } \subset B\)  that is orthogonal to \(a\)  and disjoint from \(b\) . Similarly, one can then find a disk \(a_{\perp } \subset A\)  that is dual to \(b\)  and is disjoint from both \(a\)  and \(b_{\perp }\) . There is then an obvious homeomorphism \((S^3, T) \rightarrow (S^3, T_3)\)  (well-defined up to the 2-strand braid group of the torus) that takes the four disks \(a, b_{\perp }, b, a_{\perp }\)  to, respectively \(a_1, b_1, b_3, a_3\) . The Powell class of this homeomorphism then coincides both with that determined by \(a\)  and that determined by \(b\) .
There remains the possibility that one or both primitive disks are surrogates of \(a\)  or \(b\)  and intersect, in which case they are orthogonal. There is then a homeomorphism \((S^3, T) \rightarrow (S^3, T_3)\)  that carries one disk to \(a_1\)  and the other one to \(b_1\) . But a further Powell move (an exchange) carries the pair to \(a_3\)  and \(b_3\) . The first homeomorphism is in the Powell equivalence class determined by \(a\)  or its conjugate and the second by \(b\)  or its conjugate. Since there is a Powell move taking one to the other, the equivalence classes coincide.
The second and final step is to show that any two vertices in \(2C(T)\)  that are connected by an edge determine the same Powell equivalence class. Given the simple rule above for assigning a Powell equivalence class, this will follow immediately from the following lemma.
Lemma 5.2  For the genus 3 Heegaard splitting \(S^3 = A \cup _T B\) , suppose \(a \subset A\) , \(b_1, b_2 \subset B\)  are three pairwise disjoint essential disks . Then either

\(a\)  is primitive

\(a\)  is separating and its surrogate is primitive

\(b_1\)  and \(b_2\)  are parallel in \(B\)  or

\(b_1\)  is the surrogate of \(b_2\)  or vice versa.

The symmetric statement is true for three pairwise disjoint essential disks \((a_1, a_2, b)\)

Case 1: \(a\)  is separating. If the boundary of either \(b_i\)  lies on the same side of \(a\)  as \(a^{\prime }\)  then the disk is either orthogonal to \(a^{\prime }\)  or the disk together with \(a\)  is a reducing sphere cutting off a genus 1 summand with \(a^{\prime }\)  as meridian. In any case, \(a^{\prime }\)  is primitive and we are done. So we henceforth assume that the boundary of both \(b_i\)  lie on the genus 2 side of \(T - \mbox{$\partial $}a\) , and neither is parallel to \(\mbox{$\partial $}a\) .
If, together, \(b_1\)  and \(b_2\)  separate \(T\)  then the component containing \(\mbox{$\partial $}a\)  is either a torus in which \(a^{\prime }\)  is a meridian, and so is primitive, or a genus 2 surface. In the latter case, the other component must be attached either by a single disk that cuts off a torus with the other disk as meridian (so one of \(b_1, b_2\)  is surrogate to the other) or attached by two disks, one copy of each \(b_i\) , that cut off a sphere, in which case \(b_1\)  and \(b_2\)  are parallel in \(B\) .
Case 2: \(a\)  is non-separating. One of the components of \(B - (b_1 \cup b_2)\)  contains \(\mbox{$\partial $}a\)  and has connected boundary; attaching a 2-handle to that component along \(a\)  creates a 3-manifold \(N\)  that will still be connected.
Also \(\mbox{$\partial $}N\)  is still connected, for if it were disconnected, with boundary components \(\mbox{$\partial $}_1 N, \mbox{$\partial $}_2 N\) , each containing a copy of \(a\) , then there would be a path through that component of \(B - (b_1 \cup b_2)\)  from \(\mbox{$\partial $}_1 N\)  to \(\mbox{$\partial $}_2 N\) . But there is already a path from \(\mbox{$\partial $}_1 N\)  to \(\mbox{$\partial $}_2 N\)  in \(T - a\) , since \(a\)  is non-separating. Together the paths would give a circle that intersects the surface \(\mbox{$\partial $}_1 N\)  in a single point, which is impossible in \(S^3\) .
So let \(\gamma \subset \mbox{$\partial $}N\)  be a path from one copy of \(a\)  in \(\mbox{$\partial $}N\)  to the other. Tubing the copies of \(a\)  together along the boundary of a neighborhood of \(\gamma \)  gives a separating disk \(a^+\)  in \(A\) , disjoint from the \(b_i\) , and \(a\)  is the surrogate for \(a^+\) . Now apply case 1 to the three disks \(a^+, b_1, b_2\) .
Corollary 5.3 The Powell Conjecture is true for genus 3 splittings.

Suppose two vertices in \(2C(T)\)  are connected by an edge. With no loss of generality, the two vertices are \((a, b_1)\)  and \((a, b_2)\)  with the three disks \(a, b_1, b_2\)  pairwise disjoint. In each case given by Lemma REF  the procedure described for assigning a Powell equivalence class assigns the same equivalence class to both cases.

Concluding remarks
The methods here seem to hold great promise for the general case, genus \(g \ge 4\) . But there are many difficulties: Presented with a pair of weakly reducing disks \(a \subset A, b \subset B\) , the methods of Casson-Gordon ([4]}) describe how to proceed to exhibit a complete non-separating weakly reducing collection of disks \(a_1, ..., a_{g_1}, b_{g_1 + 1}, ..., b_g\)  for \(T\) . But many choices are involved in creating such a collection, and it is not clear that different choices will lead, via Theorem REF , to the same Powell equivalence class of homeomorophisms to \(T_g\) , even under strong inductive assumptions.
To begin with, the first step in the Casson-Gordon recipe, following the choice of \(a, b\) , is to use the pair to maximally weakly reduce the splitting. But this involves choices: for example, one could choose first to expand \(a\)  to a maximal collection of compressing disks in \(A\)  that are disjoint from \(b\) . (Or one could do the reverse!) The resulting compression body lying in \(A\)  may itself be well-defined, via the arguments of [5]}, but there are many possible families of compressing disks for the compression body, and, to apply Theorem REF , a specific collection of compressing disks in \(A\)  is required.
The next step in [4]} is to maximally compress the surface \(T_{A_b}\)  just defined towards the side that contains \(B\) . The same difficulty arises: the actual compression disks into the \(B\)  side that one uses involves a choice. On the positive side, the resulting surface \(T_{BA_b}\) , which is well-defined, can be shown to bound a handlebody \(M_{BA_b}\)  in which \(T_{A_b}\)  is a Heegaard surface, providing perhaps an inductive foothold. Furthermore, Powell-like moves (bubble-moves, eyeglass twists, etc) in the Heegaard surface of \(T_{A_b} \subset M_{BA_b}\)  can be 'shadowed' by similar moves in the original splitting surface \(T\) . But again choice is involved in how to shadow the moves. Also, even once a primitive disk is chosen for the splitting \(T_{A_b} \subset M_{BA_b}\)  (inductively we might assume that this choice is unique up to Powell-like moves), such a disk does not pick out a unique primitive disk in our original surface \(T\) .
All along one could hope to show that different choices made lead to Powell equivalent trivializations, but we have failed to find such arguments.
Perhaps a good intermediate question is this:
Conjecture 6.1 Any eyeglass twist on \(T_g\)  is a Powell move.

Of course any eyeglass twist is a Goeritz element, so this conjecture is weaker than the Powell Conjecture. An affirmative solution would bode well for an affirmative solution of the Powell Conjecture as well. A weaker but perhaps still challenging conjecture is this:
Conjecture 6.2 Any eyeglass twist on \(T_g\)  is Goeritz conjugate to a Powell move.

That is, given an eyeglass twist \(\rho : (S^3, T_g) \rightarrow (S^3, T_g)\)  there is a homeomorphism \(h: (S^3, T_g) \rightarrow (S^3, T_g)\)  so that \(h^{-1}\rho h: (S^3, T_g) \rightarrow (S^3, T_g)\)  is a Powell move.
Observe the following cautionary example about eyeglass twists: In the argument of Section REF  a crucial weakly reducing pair that may well appear can be viewed as follows: One of the pair, \(b \subset B\)  say, is an innermost disk of \(B \cap S\)  for some level sphere \(S\) ; \(b\)  lies inside another disk \(D\)  that is also bounded by a curve of \(S \cap T\) , a curve to which \(b\)  is tangent. Finally, the complement of \(b\)  in \(D\)  lies entirely in \(A\)  and becomes the disk \(a\)  when the point of tangency is resolved. See Figure REF .
<FIGURE>Now imagine rolling the disk \(b\)  around the inside of \(D\)  so that the point of tangency circumnavigates \(\mbox{$\partial $}D\) . It is easy to see that this is a Goeritz move and only a bit harder to see that it is an eyeglass twist of \(b\)  around \(a\) . Yet, the methods of Section REF  would provide the same pair of disks \(a, b\)  throughout the Goeritz move. This suggests that, as it stands, Section REF  is not in itself powerful enough to resolve even the weaker Conjecture REF . Other tools may be needed to solve the problem for at least some types of eyeglass twists.
In that spirit, we conclude with one rather limited class of eyeglass twists which we are able to show are Powell moves. These are described using the notation that precedes Lemma REF .
Definition 6.3  An eyeglass \(\eta \subset T\)  satisfying the following conditions will be called a short eyeglass.

There is a complete non-separating pair of weakly reducing disk collections \(\lbrace a_1, ..., a_{g_1}\rbrace \)  and \(\lbrace b_{g_1+1}, ..., b_g\rbrace \)  for \(T \subset S^3\) .

There is a simple closed curve \(c \subset T\)  that separates the two sets \(\lbrace \mbox{$\partial $}a_1, ..., \mbox{$\partial $}a_{g_1}\rbrace \)  and \(\lbrace \mbox{$\partial $}b_{g_1+1}, ..., \mbox{$\partial $}b_g\rbrace \)  in \(T\) .

The lens \(a \subset A\)  of \(\eta \)  has \(\mbox{$\partial $}a \subset T_A\) .

The lens \(b \subset B\)  of \(\eta \)  has \(\mbox{$\partial $}b \subset T_B\) .

The interior of the bridge \(v \subset \eta \)  intersects at least one of \(T_A\)  or \(T_B\)  entirely in \(P_A\)  or \(P_B\) .


Proposition 6.4  Suppose \(\eta \subset T\)  is a short eyeglass as above, with \(c \subset T\)  the separating curve. Then an eyeglass twist along \(\eta \)  does not change the Powell equivalence class of any homeomorphism \(h:(S^3, T, c) \rightarrow (S^3, T_g, c_{g_1})\) .

The central difference between this lemma and Lemma REF  is that we no longer require the bridge \(v\)  to intersect \(c\)  in exactly one point; on the other hand, we do require that the bridge of the eyeglass be disjoint either from \(\lbrace \mbox{$\partial $}a_1, ..., \mbox{$\partial $}a_{g_1}\rbrace \)  or from \(\lbrace \mbox{$\partial $}b_{g_1+1}, ..., \mbox{$\partial $}b_g\rbrace \)
The proof is by induction on \(|v \cap c|\) ; the case \(|v \cap c| = 1\)  is Lemma REF .
Suppose with no loss of generality that \(v \cap T_A \subset P_A\)  and that \(|v \cap c| > 1\) . Consider the arcs of \(v \cap P_A\)  that are not incident to \(\mbox{$\partial $}a\) . Each such divides \(P_A\)  into two sub-planar surfaces, one of which cuts off a subarc of \(c\)  intersecting \(v\)  in an even number of points. By picking an outermost such arc we may find a properly embedded segment \(v_A \subset v\)  that cuts off a planar surface whose interior is disjoint from \(v\) . The union of \(v_A\)  and the subarc \(c_A \subset c\)  it cuts off is a simple closed curve \(a_c\)  which bounds a disk in \(A\) . Create an eyeglass \(\eta ^{\prime }\) , one of whose lenses is \(a_c\) , the other is \(b \subset B\)  and whose bridge intersects \(c\)  in at least two fewer points. By induction, a twist around \(\eta ^{\prime }\)  does not change the Powell equivalence class of \(h\) , and it vacates entirely everything in \(P_A\)  that lies between \(v_A\)  and \(c_A\) , putting it on the other side of \(v_A\) . Then \(v_A\)  can be isotoped across \(c\) , reducing \(|v \cap c|\) .

appendix

Refining the Rieck argument
We begin with a well-known result. Suppose \(X\)  is a compact topological space and \(f: X \rightarrow \mbox{$\mathbb {R} $}\)  is continuous. Then \(f\)  has a unique minimal value \(r \in \mbox{$\mathbb {R} $}\) . Now let \(Z\)  be a topological space, \(\mbox{$\mathbb {R} $}^X\)  denote the space of continuous functions from \(X\)  to \(\mbox{$\mathbb {R} $}\)  with the compact-open topology and consider a function \(f: Z \rightarrow \mbox{$\mathbb {R} $}^X\) . Denote the image of any \(z \in Z\)  by \(f_z: X \rightarrow \mbox{$\mathbb {R} $}\) . Then \(f\)  gives rise to a function \(r: Z \rightarrow \mbox{$\mathbb {R} $}\)  by defining \(r(z) = min\lbrace f_z(x) | x \in X\rbrace \) .
Proposition 1.1 If \(f\)  is continuous then so is \(r\) .

The standard proof is this: Since \(X\)  is compact, the compact-open topology coincides with the topology of uniform convergence, in which the conclusion is obvious. There is also a proof directly from the definition of compact-open topology that is analogous to (but much easier than) the proof of Proposition REF  below.
Notice that the function \(r\)  could equivalently (but more obscurely) be defined this way: \(r(z)\)  is the least \(r \in \mbox{$\mathbb {R} $}\)  so that the image of \(H_0(f_z^{-1}(- \infty , r]) \rightarrow H_0(X)\)  is non-trivial. Viewed in this way, the proposition generalizes. In particular, we will be interested in an analogous statement for \(H_1\) .
For this version, assume also that \(H_1(X)\)  is non-trivial. Then given a continuous
function \(f: X \rightarrow \mbox{$\mathbb {R} $}\)  and a value \(y \in \mbox{$\mathbb {R} $}\) , let \(G_y \subset H_1(X)\)  be the image under the inclusion-induced map of \(H_1(f^{-1}(-\infty ,y]) \rightarrow H_1(X)\) . There are obvious properties:

for \(y\)  below the minimum of \(f(X)\) , \(G_y\)  is trivial,

\(y \le y^{\prime } \Rightarrow G_y \subset G_{y^{\prime }}\) , and

for \(y\)  above the maximum of \(f(X)\) , \(G_y = H_1(X) \ne 0\) .

So it makes sense in this situation to define a value
\(r = \inf \lbrace y \in \mbox{$\mathbb {R} $}| G_y \ne 0\rbrace .\footnote {Notice that G_r may or may not be trivial. For an example where G_r is trivial, let X be the closure of an \epsilon -neighborhood of the Polish circle C in the plane, and f(x) be the distance from x to C. Then r = 0 and G_r = H_1(C) = 0.}\) 
With this definition then, in analogy to the discussion above, a function \(f: Z \rightarrow \mbox{$\mathbb {R} $}^X\)  gives rise to a function \(r: Z \rightarrow \mbox{$\mathbb {R} $}\)  by setting \(r(z)\)  to be the value \(r\)  just defined for the function \(f_z\) . That is,
\(r(z) = \inf \lbrace y \in \mbox{$\mathbb {R} $}| image(H_1(f_z^{-1}(-\infty ,y]) \rightarrow H_1(X)) \ne 0\rbrace \) 
Proposition 1.2 
Since \(f\)  is continuous, so is \(r\) .

The proof (not the underlying mathematics) is made a bit more complicated because we we do not know when \(G_r\)  is trivial.
Pick any \(z_0 \in Z\)  and any \(\mbox{$\epsilon $}> 0\) . Let \(r = r(z_0)\) .
Claim 1: There is an open neighborhood \(N\)  of \(z_0\)  so that \(z \in N \Rightarrow r(z) < r + 2\mbox{$\epsilon $}\) .
Proof of Claim 1: First observe that \(f_{z_0}^{-1}(- \infty , r + \mbox{$\epsilon $}]\)  is closed in \(X\) , hence compact, so \(\lbrace z \in Z| f_z(f_{z_0}^{-1}(- \infty , r + \mbox{$\epsilon $}]) \subset (-\infty , r + 2\mbox{$\epsilon $})\rbrace \)  is an open neighborhood \(N\)  of \(z_0\) . Put another way, for \(z \in N\) ,
\( f_{z_0}^{-1}(- \infty , r + \mbox{$\epsilon $}] \subset f_{z}^{-1}(- \infty , r + 2\mbox{$\epsilon $}).\) 
We know that the image of \(H_1(f_{z_0}^{-1}(- \infty , r + \mbox{$\epsilon $}] ) \rightarrow H_1(X)\)  is non-trivial, and this factors through \(H_1(f_{z}^{-1}(- \infty , r + 2\mbox{$\epsilon $})) \rightarrow H_1(X)\) , so the image of the latter is non-trivial. Hence \(r(z) < r + 2\mbox{$\epsilon $}\)  as required.
Claim 2: There is an open neighborhood \(M\)  of \(z_0\)  so that \(z \in M \Rightarrow r(z) > r - 2\mbox{$\epsilon $}\) .
Proof of Claim 2: Paralleling the argument in Claim 1, there is an open neighborhood \(M\)  of \(z_0\)  so that for \(z \in M\) ,
\(X - f_{z_0}^{-1}(- \infty , r - \mbox{$\epsilon $}) = f_{z_0}^{-1}[r - \mbox{$\epsilon $}, \infty ) \subset f_{z}^{-1}(r- 2\mbox{$\epsilon $}, \infty ) = X - f_{z}^{-1}(-\infty , r - 2\mbox{$\epsilon $}].\) 
Hence
\(f_{z}^{-1}(-\infty , r - 2\mbox{$\epsilon $}] \subset f_{z_0}^{-1}(- \infty , r - \mbox{$\epsilon $})\) 
By definition of \(r\) , \(H_1(f_{z_0}^{-1}(- \infty , r - \mbox{$\epsilon $})) \rightarrow H_1(X)\)  has trivial image; it follows that \(H_1(f_{z}^{-1}(- \infty , r- 2\mbox{$\epsilon $}]) \rightarrow H_1(X)\)  has trivial image, so \(r(z) > r - 2\mbox{$\epsilon $}\)  as required.
Hence for \(z \in M \cap N\) , \(|r(z) - r(z_0)| < 2\mbox{$\epsilon $}\) , showing that \(r\)  is continuous at \(z_0\) .
Now return to the setting of Section REF : \(T\)  is a genus \(g \ge 2\)  Heegaard surface in \(S^3\)  and \(T_t, t \in [-1, 1]\)  describes a sweep-out of \(S^3 = A \cup _T B\)  from a spine of \(A\)  to a spine of \(B\) . Similarly \(S_s, s \in [-1, 1]\)  describes the sweep-out of \(S^3\)  by 2-spheres, from the south pole no the north pole of \(S^3\) . The intersection patterns of the two surfaces give rise to a reduced graphic \(\Gamma \subset [-1, 1] \times [-1, 1]\)  for which a complementary region \(R\)  is labelled \(E\)  if and only if for each \((t, s) \in R\) , \(T_t \cap S_s\)  contains a circle that is essential in \(T_t\) .
Proposition 1.3 
There is a function \(r: [-1, 1] \rightarrow [-1, 1]\)  whose graph \(\lbrace (t, r(t))\rbrace  \subset [-1, 1] \times [-1, 1]\)  is transverse to \(\Gamma \)  and each region that the graph intersects is labelled \(E\) .

Apply Proposition REF  to \(X = T\)  and \(Z = [-1, 1]\)  by taking for \(f_t\)  the height function \(s\)  restricted to \(T_t \subset S^3\) . Denote by \(T_{t, (-\infty s]}\)  the subsurface of \(T_t\)  that lies in \(\cup _{s^{\prime } \le s} S_{s^{\prime }}.\) , that is, in the part of \(S^3\)  that is south of the sphere \(S_s\) . We then have a function \(r:[-1, 1] \rightarrow [-1, 1]\)  with the property that for values of \(s < r(t)\) , none of the first homology of \(T_{t, (-\infty , s]}\)  persists into all of \(T_t\) , but for any values of \(s > r(t)\)  some does. It follows that the topology of \(T_{t, (-\infty s]}\)  changes as \(s\)  rises through \(r(t)\) , so the graph of \(r(t)\)  must lie in the 1-skeleton of the reduced graphic.
We can say more: For a generic value of \(t\) , the point \((t, r(t))\)  will lie in an edge of the reduced graphic, so at that point there is a saddle tangency of \(T_t\)  with \(S_{r(t)}\) . That is, \(T_{t, (-\infty , r(t) + \mbox{$\epsilon $}]}\)  is obtained from \(T_{t,(-\infty , r(t) - \mbox{$\epsilon $}]}\)  by attaching a 1-handle.
At a non-generic value of \(t\) , the point \((t, r(t))\)  will be a vertex in the graphic. In that case, one can get from the region just below \((t, r(t))\)  to the region just above by passing through two edges in the graphic, so \(T_{t, (-\infty , r(t) + \mbox{$\epsilon $}]}\)  is obtained from \(T_{t, (-\infty , r(t) - \mbox{$\epsilon $}]}\)  by attaching two 1-handles. By definition of \(r(t)\) , \(T_{t, (-\infty , r(t) - \mbox{$\epsilon $}]}\)  corresponds to a subsurface of \(T_t\)  whose first homology is trivial in all of \(T_t\) . In particular, \(T_{t, (-\infty , r(t) + \mbox{$\epsilon $}]}\)  contains no non-separating simple closed curves i. e., \(T_{t, (-\infty , r(t) + \mbox{$\epsilon $}]}\)  is planar.
We now invoke an easy lemma:
Lemma 1.4
Suppose \(P\)  is a compact possibly disconnected planar surface and \(P_+\)  is the surface obtained from \(P\)  by attaching no more than three 1-handles. Then either genus \(P_+ \le 1\)  or some boundary component of \(P\)  is non-separating in \(P_+\) .

If any of the 1-handles has its ends attached to different components of \(P\) , it is wasted: The resulting surface \(P^{\prime }\)  is still planar, and the new boundary component created is non-separating in \(P_+\)  if and only if at least one of the boundary components to which it is attached is non-separating in \(P_+\) . So we could juat replace \(P\)  with \(P^{\prime }\)  in the argument, but now with one fewer handle to attach.
So we may as well assume that all 1-handles are attached to the same component of \(P\)  and so we can take \(P\)  to be connected. In that case, if any 1-handle has exactly one end on any component of \(\mbox{$\partial $}P\)  then that component becomes non-separating, finishing the argument. So we may as well assume that no 1-handle has a single end on an original boundary component of \(P\) .
Now imagine attaching the 1-handles sequentially, say \(h_1, h_2, h_3\) . The first handle \(h_1\)  must have both ends on the same component of \(\mbox{$\partial $}P\)  so the result is still a planar surface \(P^{\prime }\) , with exactly two components of \(\mbox{$\partial $}P^{\prime }\)  not appearing as components of \(\mbox{$\partial $}P\) . If, after attaching \(h_2\) , the surface is still planar, attaching \(h_3\)  can raise the genus to at most 1, completing the proof. So we need to have the attachment of \(h_2\)  raise the genus, so it must connect the two new boundary components of \(P^{\prime }\) . The resulting surface \(P^{\prime \prime }\)  now has genus 1 and only a single boundary component not among the components of \(\mbox{$\partial $}P\) . It follows that both ends of \(h_3\)  must lie on the same component of \(\mbox{$\partial $}P^{\prime \prime }\) , either the new one or one of the original components of \(\mbox{$\partial $}P\) . In any case, attaching \(h_3\)  does not raise the genus any further, completing the proof.
Following Lemma REF , if every boundary component of \(T_{t, (-\infty , r(t) + \mbox{$\epsilon $}]}\)  is inessential in \(T_t\)  then \(genus(T_t) \le 1.\)  So under the hypothesis that \(genus(T) \ge 2\)  the function \(r(t) + \mbox{$\epsilon $}\)  for small enough \(\mbox{$\epsilon $}\)  will be the function we seek: its graph in \([-1, 1] \times [-1, 1]\)  will lie entirely in the regions labelled \(E\) .
The argument is readily enlarged to include the third parameter \(\theta \) . That is,
Proposition 1.5  In the parameter space \([-1, 1] \times [-1, 1]\times S^1\)  as described in Section REF , there is a function \(r: I \times S^1 \rightarrow I\)  so that the graph \((t, r(t, \theta ) + \mbox{$\epsilon $}, \theta ) \subset [-1, 1] \times [-1, 1] \times S^1\)  of \(r + \epsilon \)  is transverse to the graphic and intersects only those regions of the graphic in which \(S_s \cap T_{t, \theta }\)  contains circles that are essential in \(T_{t, \theta }\) .

The only additional case to consider is that in which \((t, r(t, \theta ), \theta )\)  is a (codimension 3) vertex in the graphic. But in that case, the region just below the vertex can be connected to the region just above the vertex by passing through three saddle “walls". Thus Lemma REF  is still applicable, now with \(n = 3\) .
